To connect ideas 
between   sentences, the following adverbs can be used:
however, nevertheless, on the other hand, otherwise, though, still, all the same.
 
The more he talked the more silent Mary became. However   he thought she had been interested.
To connect ideas 
within   sentences the following adverbs can be used:
although, though, even though, yet, while, in spite of / despite +noun, 
in spite of / despite+noun or despite+verb(ing).